Questions: Family & Friends

I'm 18 and currently living with my parents (I graduated last June and have trouble finding a job). In that house also lives my older brother (21), my younger sister (16), and the sister in question (14).

So my sister is 14 and got kicked out of school for poor performance (35%! She skipped almost every day). The only thing she does is staying at home playing video games. As both my mother and brother have to work late, I offered to take care of the house.

Well, my duties is keeping the house clean and cook for my sisters. Since I don't see why I should be the only one cleaning, I give menial tasks to my sisters (empty the dish washer, take out the trash etc.).

Yesterday I asked my 14-year-old sister to clean a dirty window. She was busy playing GTA and told me "later". 2 hours later after I was done with everything else I asked her if she would EVER do the window. "Goddamnit, later" was all I got back. I cleaned it myself and it took me 2 minutes. After I was done I told her that doing nothing is what a useless brat like her does best. She got angry and said "I fucking hate you, go fuck yourself" while she ran to her room. I followed her and told her that she just can't talk to me like that. She then showed her middle finger to me and said "fuck you". That's when I exploded in rage and slapped her in the face. It was probably very hard, as she was crying and holding her left eye and cheek.

Was I wrong to slap her?

And how could I have handled it better?